Heat conduction and sound transmission in isotopically disordered harmonic crystals

We investigate some kinetic properties of an isotopically disordered harmonic crystal. We prove rigorously that for almost all disordered chains the transmission coefficient of a plane wave with frequency ω, t N(ω), decays exponentially in N, the length of the disordered chain, with the decay constant proportional to ω² for small ω. The response of this system to an incident wave is related to the nature of the heat flux J(N) in a disordered chain of length N placed between heat reservoirs whose temperatures differ by ΔT>0. We clarify the relationship between the works of various authors in the heat conduction problem and establish that for all models J(N)→0 as N→∞ in a disordered system. The exact asymptotic dependence of J(N) on N eludes us, however. We also investigate the heat flow in a simple stochastic model for which Fourier's law is shown to hold. Similar results are proven for two-dimensional systems disordered in one direction.